Mathematics for the interested outsider

Representations of Hopf Algebras I

We’ve seen that the category of representations of a bialgebra is monoidal. What do we get for Hopf algebras? What does an antipode buy us? Duals! At least when we restrict to finite-dimensional representations.

Again, we base things on the underlying category of vector spaces. Given a representation , we want to find a representation . And it should commute with the natural transformations which make up the dual structure.

Easy enough! We just take the dual of each map to find . But no, this can’t work. Duality reverses the order of composition. We need an antiautomorphism to reverse the multiplication on . Then we can define .

The antiautomorphism we’ll use will be the antipode. Now to make these representations actual duals, we’ll need natural transformations and . This natural transformation is not to be confused with the counit of the Hopf algebra. Given a representation on the finite-dimensional vector space , we’ll just use the and that come from the duality on the category of finite-dimensional vector spaces.

Thus we find that is the pairing . Does this commute with the actions of ? On the one side, we calculate

Then we apply the evaluation to find

Which is the same as the result we’d get by applying the “unit” action after evaluating. Notice how we used the definition of the dual map, the fact that is a representation, and the antipodal property in obtaining this result.

This much works whether or not is a finite-dimensional vector space. The other direction, though, needs more work, especially since I waved my hands at it when I used as the motivating example of a category with duals. Tomorrow I’ll define this map.

Share this:

Like this:

Related

[…] Representations of Hopf Algebras II Now that we have a coevaluation for vector spaces, let’s make sure that it intertwines the actions of a Hopf algebra. Then we can finish showing that the category of representations of a Hopf algebra has duals. […]

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.